论文标题
中间第三个插座设置的近似元素，带有二元理性
Approximating elements of the middle third Cantor set with dyadic rationals
论文作者
论文摘要
令$ c $为中间第三个康托尔集，$μ$为$ \ frac {\ log 2} {\ log 3} $ - 尺寸hausdorff量度仅限于$ c $。在本文中，我们研究了二元理性的$ c $元素的近似值。我们的主要结果意味着，对于$μ$，几乎每$ x \ c $中的每一个$ x \我们都有$ \＃\ weft \ {1 \ leq n \ leq n \ leq n：\ left | x- \ frac {p} {2^n} {2^n} \ right | \ leq \ frac {1} {n^{0.01} \ cdot 2^{n}}} \ textrm {对于某些} p \ in \ Mathbb {n} \ right \} \ right \} \ sim 2 \ sum 2 \ sum_________ { YU对微不足道的近似率提供了亚属性的改进。
Let $C$ be the middle third Cantor set and $μ$ be the $\frac{\log 2}{\log 3}$-dimensional Hausdorff measure restricted to $C$. In this paper we study approximations of elements of $C$ by dyadic rationals. Our main result implies that for $μ$ almost every $x\in C$ we have $$\#\left\{1\leq n\leq N:\left|x-\frac{p}{2^n}\right| \leq \frac{1}{n^{0.01}\cdot 2^{n}}\textrm{ for some }p\in\mathbb{N}\right\}\sim 2\sum_{n=1}^{N}n^{-0.01}.$$ This improves upon a recent result of Allen, Chow, and Yu which gives a sub-logarithmic improvement over the trivial approximation rate.
